2015 AMC 10B Problem 21
Problem 21 of 25HarderAlgebraNumber Theory
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump.
Cozy goes two steps up with each jump (though if necessary, he will just jump the last step).
Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than steps left).
Suppose Dash takes fewer jumps than Cozy to reach the top of the staircase. Let denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of
Answer choices
Show solution
Solution
Suppose Dash takes jumps. Then the number of steps is one of Cozy takes more jumps, so Cozy takes jumps, which means is either or .
Matching these possibilities, the integer solutions are They give , respectively.
Thus , and the sum of its digits is .
Thus, the correct answer is D.